Question: Mr. Sanchez's students were asked to add two positive integers. Juan subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the correct answer?
Answer: Call the two positive integers $x$ and $y$. Without loss of generality, assume $x > y$. We can write a system of equations to represent the information given in the problem:  \begin{align*}
x - y &= 2 \\
x \cdot y &= 120
\end{align*} Solving for $x$ in the first equation yields $x = y + 2$.

Substituting this into the second equation gives $(y + 2) \cdot y = 120$, or $y^2 + 2y - 120 = 0$.

This quadratic equation factors into $(y + 12)(y-10) = 0$, so $y = 10$.

Given $y$, we can solve for $x$ to get $x = 12$, so $x + y = \boxed{22}$.